Evaluate each function at the given values of the independent variable and simplify. h(x) = x³ − x + 1 d. h (3a)

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with evaluating the function h(x) = x³ − x + 1 at the given value of the independent variable, which is h(3a). This means substituting 3a for x in the function.
2. Step 2: Substitute 3a into the function h(x). Replace every occurrence of x in the function with 3a. The function becomes h(3a) = (3a)³ − (3a) + 1.
3. Step 3: Simplify the first term (3a)³. Use the property of exponents: (3a)³ = 3³ * a³ = 27a³.
4. Step 4: Simplify the second term −(3a). This simplifies directly to −3a.
5. Step 5: Combine all terms to express h(3a) in its simplified form. The result is h(3a) = 27a³ − 3a + 1.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value for the independent variable in a function. In this case, to evaluate h(3a), we replace x in the function h(x) = x³ − x + 1 with 3a. This process allows us to find the output of the function for that particular input.

Polynomial Functions

A polynomial function is a mathematical expression that involves variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function h(x) = x³ − x + 1 is a polynomial of degree three, which means its highest exponent is three. Understanding the structure of polynomial functions is essential for evaluating and simplifying them.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form by combining like terms and performing arithmetic operations. After substituting 3a into the function, we will need to simplify the resulting expression, which may involve expanding terms and collecting similar components to achieve a clearer representation of the output.